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12.2.2.8: Oblique Shock Examples

Example 12.3

Air flows at Mach number (\(M_1\)) or \(M_{x} = 4\) is approaching a wedge. What is the maximum wedge angle at which the oblique shock can occur? If the wedge angle is \(20^\circ\), calculate the weak, the strong Mach numbers, and the respective shock angles.

Solution 12.3

The maximum wedge angle for (\(M_x= 4\)) \(D\) has to be equal to zero. The wedge angle that satisfies this requirement is by equation (??) (a side to the case proximity of \(\delta=0\)). The maximum values are:

Oblique Shock Input: \(M_x\) k = 1.4
\(M_x\) \(M_y\) \(\delta_{max}\) \(\theta{max}\)
4.000 0.97234 38.7738 66.0407

To obtain the results of the weak and the strong solutions either utilize the equation (28) or the GDC which yields the following results

Oblique Shock Input: \(M_x\) k = 1.4
\(M_x\) \({{M_y}_s}\) \({{M_y}_w}\) \(\theta_{s}\) \(\theta_{w}\) \(\delta\)
4.000 0.48523 2.5686 1.4635 0.56660 0.34907

Fig. 12.11 Oblique shock occurs around a cone. This photo is courtesy of Dr.~Grigory Toker, a Research Professor at Cuernavaco University of Mexico. According to his measurement, the cone half angle is \(15^\circ\) and the Mach number is 2.2.

Example 12.4

A cone shown in Figure 12.11 is exposed to supersonic flow and create an oblique shock. Is the shock shown in the photo weak or strong shock? Explain. Using the geometry provided in the photo, predict at which Mach number was the photo taken based on the assumption that the cone is a wedge.

Solution 12.4

The measurements show that cone angle is \(14.43^\circ\) and the shock angle is \(30.099^\circ\). With given two angles the solution can be obtained by utilizing equation (59) or the Potto-GDC.

Oblique Shock Input: \(M_1\) k = 1.4  
\(M_1\) \({{M_y}_s}\) \({{M_y}_w}\) \(\theta_{s}\) \(\theta_{w}\) \(\delta\) \(\dfrac{{P_0}_y}{{P_0}_x}\)
3.2318 0.56543 2.4522 71.0143 30.0990 14.4300 0.88737

Because the flow is around the cone it must be a weak shock. Even if the cone was a wedge, the shock would be weak because the maximum (transition to a strong shock) occurs at about \(60^{\circ}\). Note that the Mach number is larger than the one predicted by the wedge.

Fig. 12.12 Maximum values of the properties in an oblique shock.

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.